A thermodynamics problem for my friend Swapnil

Classical Mechanics Level 5

A bubble of radius \(r=5\text{ cm}\) containing a diatomic ideal gas, has the soap film of thickness \(h= 10 \ \mu \text{m}\) and is placed in vacuum. The soap film has the surface tension \( \sigma = 0.04\text{ N m}^{-1}\) and the density \( \rho =1.1 \text{ g cm}^{-3}\).

  1. Find formula for the molar heat capacity of the gas in the bubble for such a process when the gas is heated so slowly that the bubble remains in a mechanical equilibrium and evaluate it. Let the answer be \(x\).
  2. Find formula for the frequency \( \omega \) of the small radial oscillations of the bubble and evaluate it under the assumption that the heat capacity of the soap film is much greater than the heat capacity of the gas in the bubble. Assume that the thermal equilibrium inside the bubble is reached much faster than the period of oscillations. Let the answer be \(y\).

Compute your answer as \( x+y \) upto 1 decimal place.


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